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When does inferring reputation probability countervail temptation in cooperative behaviors for the prisoners’ dilemma game?
Chaos, Solitons and Fractals 78 (2015) 238–244
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
When does inferring reputation probability countervail temptation in cooperative behaviors for the prisoners’ dilemma game? Yu Dai a, Peng Lu a,b,∗ a b
Department of Sociology, Tsinghua University, Beijing, China Department of Automation, Tsinghua University, Beijing, China
a r t i c l e
i n f o
Article history: Received 30 April 2015 Accepted 29 July 2015 Available online 1 September 2015 Keywords: Reputation Countervail Temptation Cooperation
a b s t r a c t In evolutionary games, the temptation mechanism reduces cooperation percentage while the reputation mechanism promotes it. Inferring reputation theory proposes that agent’s imitating neighbors with the highest reputation takes place with a probability. Although reputation promotes cooperation, when and how it enhances cooperation is still a question. This paper investigates the condition where the inferring reputation probability promotes cooperation. Hence, the effects of reputation and temptation on cooperation are explored under the spatial prisoners’ dilemma game, utilizing the methods of simulation and statistical analysis. Results show that temptation reduces cooperation unconditionally while reputation promotes it conditionally, i.e. reputation countervails temptation conditionally. When the inferring reputation probability is less than 0.5, reputation promotes cooperation substantially and thus countervails temptation. However, when the inferring reputation probability is larger than 0.5, its contribution to cooperation is relatively weak and cannot prevent temptation from undermining cooperation. Reputation even decreases cooperation together with temptation when the probability is higher than 0.8. It should be noticed that inferring reputation does not always succeed to countervail temptation and there is a specific interval for it to promote cooperation. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction For researchers, promoting cooperation has been a longlasting pursuit as cooperation brings public good for our society [1–9]. In evolutionary game theories, it has been substantially researched that temptation seduces individuals to defect or cheat on cooperative partners and therefore reduces the cooperation propensity or rate within most groups [1,5,10]. Temptation means that the defect strategy brings more payoffs, which suits Darwinism, and individuals tend to
∗ Corresponding author at: Department of Sociology, Tsinghua University, Beijing, China. Tel.: +86018811759063. E-mail address: [email protected] (P. Lu).
http://dx.doi.org/10.1016/j.chaos.2015.07.030 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
defect other than cooperate [5,6,10]. If no mechanisms counterbalance temptation, cooperation will be doomed. In order to enhance cooperation, related models or solutions are proposed [1–10]. In evolutionary game theories, the core idea is to design certain mechanisms countervailing temptation that undermines cooperation [6,11–13]. Three models are commonly applied to investigate temptation and possible countervailing mechanisms [14–25], such as prisoner’s dilemma game, snowdrift game and public good game. As the leading paradigm, the prisoner’s dilemma game is widely utilized [5]. Related counter-temptation mechanisms or solutions have been developed to countervail temptation and promote cooperation. Each of them captures certain patterns of human’s behavior, such as kin selection [26,27], direct reciprocity [27,28], indirect reciprocity
Y. Dai, P. Lu / Chaos, Solitons and Fractals 78 (2015) 238–244
Fig. 1. The payoff matrix. Each individual who cooperates gets one if the partner is cooperating. Each one who defects obtains b (b > 1) if the partner is cooperating. In other cases, he or she gets noting instead.
[29–32], group selection [33,34], tolerance [35], altruism punishment [36–38], spatially structured populations [8,39–44], social values [13,45], selective investment [3,9,46], social diversity [2,4,7,9,47], volunteerism [37,48–50], aspirations [51,52], multilayer networks [53–55], and complex networks [56,57]. Apart from them, the reputation mechanism is commonly introduced as a necessary countervailing mechanism against temptation [5,6,13,50,58–61]. Some researchers investigated the effects of reputation on the individual partner-switching process, and it indicates that the group’s cooperation propensity will increase by almost 100% if individuals are free to alter actions or partners [6]. However, some researchers argue that individuals’ reputation cannot be fully identified in reality, due to the cost and error of information dissemination [5,59]. Individuals have limited information and heterogeneous capabilities to identify the neighbors’ reputation correctly. Thus, the term of inferring reputation probability is applied to refer to the situation that each agent infers the neighbor∗ with highest reputation correctly merely with a probability p, which is normally distributed. Outcomes show that inferring reputation promotes cooperation more than traditional ways [5,58]. Although we have already known that temptation reduces cooperation substantially [5,6,38,58,59], two issues stay unclear: First, how different levels of p influence the cooperation propensity, i.e. p’s effect on cooperation rate; if reputation promotes cooperation, the second issue naturally emerges that when and where reputation effectively countervails temptation, i.e. the condition where reputation countervails temptation. The first question lays the foundation of the second one. This work applies spatial prisoners’ dilemma game to investigate and solve these issues. 2. Model Agents commonly play prisoners’ dilemma games on a square lattice [5,38] in existing researches. The prisoners’ dilemma game is also called as social dilemma game [5,57]. Agents play games with its eight neighbors, and each has two strategy options, which is cooperate and defect that are denoted as C and D respectively. Payoff matrix is shown in Fig. 1, which has only one parameter b that satisfies b ∈ (1, 2] [5,6]. If one cooperates with a neighbor who cooperates he receives one, if he or she defects with a neighbor who cooperates then the payoff is b, and the payoff would be zero otherwise. According to previous work [5,6], we utilize Eq. (1) to generate reputation for each agent. Initially, each agent is given a reputation of one. Afterwards, reputation is determined by the increment of reputation, or the strategy chosen by each agent i at each time t, i.e. i z. For each agent, i z is one if he or she cooperates and zero if he or she defects. The term Zi (t − 1) represents individual’s reputation at time t − 1, and
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Fig. 2. The action rule. Neighbor∗ refers to the neighbor with the highest reputation. For each agent, there is a referring reputation probability p, and each one imitates the action of neighbor∗ at the probability of p. If this agent chose to imitate neighbor∗ , the probability is Pij , i.e. the transition probability. One imitates a random neighbor otherwise.
Zi (t) is the accumulative current reputation of each one at time t.
Zi (t ) = Zi (t − 1) + i Z
(1)
Neighbor∗
denotes the neighbor with highest reputations. Each agent imitates his or her neighbor∗ with a probability p called inferring reputation probability [5]. In order to investigate effects of p, we assume that agents share the same level of p that satisfies p ∈ (0, 1]. Hence, as is indicated in Fig. 2, each agent finds his or her own neighbor∗ with the probability of p, and imitates action of the neighbor∗ with the transition probability Pij . In other cases, agent imitates one neighbor randomly. The formula of transition probability Pij is shown by Eq. (2), where si and sj represent agent’s current action and action of his or her neighbor∗ . At each time t or interaction, each agent chooses one specific action, either cooperation or defection, and receives a payoff. Terms ui and uj are payoffs agent i and j, respectively. The focal agent i adopts action of his or her neighbor∗ j with the transition probability Pij . Therefore, the higher the payoff of j is than that of i, the more possible it is for the focal agent to imitate action of j. Equivalently, the lower the payoff of j is than that of i, the less possible for i to imitate action of j [5,6,56].
Pi j = Psi →s j =
1 1 + e(u j −ui )β
(2)
In Eq. (2), β represents the intensity of selection, which means that β → 0 leads to random drift while β → ∞ deterministic imitation. Given that β is not our focus, we set β ≡ 1. We set the number of agents 40,000 (200 × 200) and the initial percentage or propensity of cooperation is 50%. The probability p takes on 10 typical values, i.e. p ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}. Likewise, b takes on 5 typical values from interval (1, 2], i.e. b ∈ {1.2, 1.4, 1.6, 1.8, 2.0}. For each combination of parameters, the simulation process runs for 800 iterations or periods and the cooperation propensity or percentage ρ c is recorded at each time or iteration t. 3. Reputation versus temptation There exist two mechanisms, temptation and reputation, that jointly determine the cooperation rate ρ c . The temptation mechanism means that how temptation affects cooperation. This mechanism produces a negative effect on cooperation propensity [5,6,38,58]. The reputation mechanism indicates reputation’s effect on cooperation, which is the focus of this work. We investigate absolute and relative effects of the two mechanisms on cooperation. The temptation
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Fig. 3. The temptation effect. It indicates the relationship between temptation b and rate of cooperation ρ c , and (a) represents the situation where p is no larger than 0.5 and (b) depicts the case where p > 0.5. In both (a) and (b), temptation always undermines cooperation.
mechanism is measured by parameter b, and a larger b represents a stronger temptation mechanism. Likewise, the reputation mechanism is measured by p; hence a higher p represents a greater reputation mechanism [5]. We utilize the averaged cooperation rate ρ c to measure the overall propensity of cooperation. Effects of these two mechanisms are shown as follows. 3.1. Temptation’s effect In order to inspect effects of temptation b, inferring reputation probability p is kept at the same levels. It is clear that the effect of temptation b on cooperation is clear and obvious at each level of p. It always undermines cooperation ρ c , which consists with previous studies [5,6,38,57]. Fig. 3 indicates that b reduces ρ c at any given levels of p, i.e. ρ c declines with a growing b, and a larger b comes up with a lower ρ c . When p is no larger than 0.5, this pattern is clear in Fig. 3(b). When p is larger than 0.5 or close to 1, we have to zoom in to find this pattern in Fig. 3(a). This pattern is because b seduces agents to defect other than to cooperate. As it shows in Fig. 1, individuals tend to defect if the payoff is larger than one (b > 1). The larger b is, the more agents tend to defect. In all, temptation permanently seduces people to defect as the reducing force of cooperation.
that reputation reduces cooperation when it grows closely to one. Fig. 4 tells us that there should be some type of threshold for the inferring reputation probability. When p is less than or equal to 0.5, the cooperation propensity increases, and it decrease gradually when p is larger than 0.5, especially when p is larger than 0.8. This pattern holds true for each levels of temptation. At early stages, reputation mechanism plays the role of improving cooperation, but it reduces cooperation when it increases to 1. 3.3. The joint effect It concludes that reputation and temptation jointly affect or determine the propensity of cooperation, ρ c . The temptation mechanism constantly reduces cooperation, while the reputation improves cooperation at the early stage and then reduces it when it grows close to 1. These two mechanisms jointly determine the propensity of cooperation. As is shown in Fig. 5, temptation always reduces cooperation, while reputation enhances it when p < 0.6, and reduces it when p is close to 1. Hence, it appears that reputation merely countervails temptation when p is relatively small and it does not counteract temptation when p is large. 4. The countervailing condition
3.2. Reputation’s effect Likewise, we investigate effects of reputation at same levels of temptation. Therefore, we inspect the relationship between ρ c and p at each given level of b. The effect of reputation is different from temptation, in that it enhances cooperation at first and then decreases it when close to 100%. Related work has found that reputation or inferring reputation increases cooperation [5,6,60], but it is not well revealed
The reputation countervails temptation merely when the inferring probability p is small but it cannot counteract temptation when it grows larger and even reduces cooperation as it is close to 100%. Therefore, the countervailing condition of reputation is that p should be less than 0.5. This condition is acquired by observations, and we ought to test it with a systemic or scientific way. Hence, we apply statistical methods to figure out the validity of this countervailing condition.
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Fig. 4. The reputation effect. The relationship between and the inferring reputation probability p is shown here, and we can see two thresholds here. The first threshold is about 0.5. Before 0.5, p enhances substantially, but after that it cannot do this anymore. The second threshold is about 0.9, after that p decreases.
Fig. 5. Joint effects of reputation and temptation. The combined effect of reputation (horizontal axis) and temptation (vertical axis) is shown here. Temptation has five typical values and reputation owns 10. For temptation, it reduces cooperation constantly, but reputation enhances cooperation conditionally. As p > 0.9, it decrease cooperation instead.
First we extend the simulation to get a much denser dataset. We run the simulation for 800 iterations with more parameters, i.e. p ∈ {0.025, 0.05, 0.075, 0.1, 0.125, 0.15, 0.175, 0.2, 0.225, 0.25, 0.275, 0.3, 0.325, 0.35, 0.375, 0.4, 0.425, 0.45, 0.475, 0.5, 0.525, 0.55, 0.575, 0.6, 0.625, 0.65, 0.675, 0.7, 0.725, 0.75, 0.775, 0.8, 0.825, 0.85, 0.875, 0.9, 0.925, 0.95, 0.975, 1} and b ∈ {1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0}. For each simulation, the average propensity of cooperation is recorded to indicate the overall cooperation level. We plot the new heat map with a denser dataset in Fig. 6, and the overall trend is similar to that of Fig. 5. In order to examine their effects, we calculate the conditional expectation for b and p and plot them in Fig. 6. In Fig. 6(a), temptation always undermines cooperation at its whole range, which coincides with previous work [5,6]. However, reputation promotes cooperation only at its first half domain, i.e. reputation seems to promote cooperation when p < 0.5. When p > 0.5, reputation does not enhance cooperation and even reduces cooperation while p → 1, which contradicts the previous research [5].
There seems to be threshold of p = 0.5, below which reputation promotes cooperation and beyond which cooperation does not increase cooperation. We apply statistical methods to investigate their effects and verify the threshold phenomena. Eq. (3) is applied to evaluate their effects on cooperation, and estimated coefficients represent effects or forces of reputation p and temptation b. Agents are influenced by both temptation and reputation when making choices and interactions. The dependent variable is ρ c ; coefficient β b measures effects of temptation; β p measures effects of reputation; C is constant and ɛ is the residual term.
ρc = C + βb b + β p p + ε
(3)
Method of OLS (Ordinary Least Square) estimation is applied to investigate their effects. Outcomes of OLS estima and β are OLS estitions are listed in Table 1, where β p b mations of coefficients β p and β b in Eq. (3). Coefficients are estimated under four conditions of inferring reputation probability, i.e. p ∈ (0, 1), (0, 0.5), (0.5, 1], and (0.8, 1]. For the effect
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Fig. 6. Denser joint effects. The combined effect of reputation (horizontal axis) and temptation (vertical axis) is shown here. Temptation has 10 typical values and reputation has 50, which is why this map is denser than Fig. 5. For temptation, it reduces cooperation constantly, but reputation enhances cooperation conditionally. As p > 0.9, it decreases cooperation instead.
Fig. 7. Conditional effects of reputation and temptation. It visualizes the effects of reputation and temptation based on different p to investigate the condition where reputation is able to overcome or conquer temptation to increase cooperation. When p < 0.5, the reputation overcomes temptation, but it does not when p > 0.5. Especially, when p > 0.8, both temptation and reputation reduce cooperation as their effects or coefficients are negative. Hence, reputation countervails temptation successfully merely when p is less than a half.
of temptation, it shows that temptation always reduces cooperation under all conditions of p: the overall effect of tempta(.|0 < p ≤ 1) = −0.32; conditional eftion is negative, i.e. β b (.|0 < p < 0.5) = fects of temptation are all negative, i.e. β b (.|0.8 < p ≤ 1) = (.|0.5 < p ≤ 1]) = −0.32, and β −0.69, β b b −0.02. For the effect of reputation, the overall effect on co (.|0 < p ≤ 1) = 0.53. However, operation is positive, i.e. β p its conditional effects vary a great deal: the effect is positive and reputation promotes cooperation as p is below the (.|0 < p < 0.5) = 1.32 but reputation does threshold, i.e. β p not promote cooperation as it goes beyond its threshold. The higher p is, the more it undermines cooperation, because of (.|0.5 < p ≤ 1) = −0.01 and β (.|0.8 < p ≤ 1) = −0.08. β p p
Table 1 Overall and conditional effects of reputation and temptation.
β p β b C N Adj. R2 ∗∗
p ∈ (0, 1]
p ∈ (0, 0.5)
p ∈ (0.5, 1]
p ∈ (0.8, 1]
0.53∗∗∗ (0.039) −0.32∗∗∗ (0.039) 1.08∗∗∗ (0.065) 200 0.5588
1.32∗∗∗ (0.104) −0.69∗∗∗ (0.047) 1.45∗∗∗ (0.078) 100 0.8083
−0.01∗ (0.005) −0.32∗∗∗ (0.003) 1.02∗∗∗ (0.006) 100 0.2454
−0.08∗∗∗ (0.018) −0.02∗∗∗ (0.004) 1.09∗∗∗ (0.017) 40 0.4238
p < 0.01. ∗ p < 0.05. ∗∗∗ p < 0.001.
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Table 1 shows OLS estimations of reputation and temp representing their ef and β tation, the two coefficients β p b fects respectively. Coefficients are all significant at the level of 0.001. Fig. 7 visualizes effects of temptation and reputation. The overall effect of reputation is positive (0.53), and effect of reputation is negative (−0.32). However, their effects vary with different ranges of p. In terms of reputation, when p < 0.5, the promoting force of it is strongest, which is 1.32. Nevertheless, it tends to reduce cooperation when p > 0.5, and the effect is -0.01. Furthermore, it reduces cooperation even more when p > 0.8. It is statistically significant that reputation promotes cooperation when its inferring probability is relatively lower, and tends to reduce cooperation when p > 0.5. In terms of temptation, its negative effect on cooperation decreases with p. The force analysis can be done as a metaphor. Two forces of reputation and temptation jointly influence individual choices of cooperation. Temptation plays the role of negative force to undermine cooperation all the time, and reputation plays the role of positive force to enhance cooperation conditionally. When inferring reputation probability is relatively weaker, the force of inferring reputation countervails temptation and successfully leads the cooperation to defeat defection. However, the situation changes, as p grows higher than its threshold 0.5. Inferring reputation does not promote cooperation, but reduces it together with temptation. The overall force of temptation and reputation is negative, which means that this joint force undermines cooperation when p is close to 1. This is contradictory to previous work and should arouse some further considerations. 5. Conclusions Inferring reputation resists temptation and promotes cooperation conditionally, which is not well revealed before. It was previously believed that reputation always promotes cooperation and therefore countervails temptation [5,6,58,60]. Nevertheless, it does not always do this as we wished. Actually, when the inferring reputation mechanism is weaker (p < 0.5), reputation prevents cooperation from being undermined by temptation and enhance cooperation substantially. However, when p is lager than 0.5, it does not countervail temptation, as it contributes little to cooperation or even reduces it. Therefore, it cannot stand that reputation always promotes cooperation. Although it has been verified that temptation reduces cooperation unconditionally [1,6,38], we still have new findings that the reducing force of temptation varies depending on the inferring probability p as well. The negative force of temptation decreases with the inferring probability. As is shown in Fig. 7, it is strong when p is smaller, but weak when p is larger. Previous studies believe that temptation reduces cooperation permanently, but they did not reveal how temptation affects cooperation with other parameters. There also exist two limitations. First, in order to investigate effects of reputation, it is assumed that homogenous agents share the same inferring reputation. Actually, agents or people are heterogeneous and p is usually normally distributed [5]. Other than normal distribution, there exist t-distribution, Poisson distribution, scale-free distribu-
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
When does inferring reputation probability countervail temptation in cooperative behaviors for the prisoners’ dilemma game? Yu Dai a, Peng Lu a,b,∗ a b
Department of Sociology, Tsinghua University, Beijing, China Department of Automation, Tsinghua University, Beijing, China
a r t i c l e
i n f o
Article history: Received 30 April 2015 Accepted 29 July 2015 Available online 1 September 2015 Keywords: Reputation Countervail Temptation Cooperation
a b s t r a c t In evolutionary games, the temptation mechanism reduces cooperation percentage while the reputation mechanism promotes it. Inferring reputation theory proposes that agent’s imitating neighbors with the highest reputation takes place with a probability. Although reputation promotes cooperation, when and how it enhances cooperation is still a question. This paper investigates the condition where the inferring reputation probability promotes cooperation. Hence, the effects of reputation and temptation on cooperation are explored under the spatial prisoners’ dilemma game, utilizing the methods of simulation and statistical analysis. Results show that temptation reduces cooperation unconditionally while reputation promotes it conditionally, i.e. reputation countervails temptation conditionally. When the inferring reputation probability is less than 0.5, reputation promotes cooperation substantially and thus countervails temptation. However, when the inferring reputation probability is larger than 0.5, its contribution to cooperation is relatively weak and cannot prevent temptation from undermining cooperation. Reputation even decreases cooperation together with temptation when the probability is higher than 0.8. It should be noticed that inferring reputation does not always succeed to countervail temptation and there is a specific interval for it to promote cooperation. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction For researchers, promoting cooperation has been a longlasting pursuit as cooperation brings public good for our society [1–9]. In evolutionary game theories, it has been substantially researched that temptation seduces individuals to defect or cheat on cooperative partners and therefore reduces the cooperation propensity or rate within most groups [1,5,10]. Temptation means that the defect strategy brings more payoffs, which suits Darwinism, and individuals tend to
∗ Corresponding author at: Department of Sociology, Tsinghua University, Beijing, China. Tel.: +86018811759063. E-mail address: [email protected] (P. Lu).
http://dx.doi.org/10.1016/j.chaos.2015.07.030 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
defect other than cooperate [5,6,10]. If no mechanisms counterbalance temptation, cooperation will be doomed. In order to enhance cooperation, related models or solutions are proposed [1–10]. In evolutionary game theories, the core idea is to design certain mechanisms countervailing temptation that undermines cooperation [6,11–13]. Three models are commonly applied to investigate temptation and possible countervailing mechanisms [14–25], such as prisoner’s dilemma game, snowdrift game and public good game. As the leading paradigm, the prisoner’s dilemma game is widely utilized [5]. Related counter-temptation mechanisms or solutions have been developed to countervail temptation and promote cooperation. Each of them captures certain patterns of human’s behavior, such as kin selection [26,27], direct reciprocity [27,28], indirect reciprocity
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Fig. 1. The payoff matrix. Each individual who cooperates gets one if the partner is cooperating. Each one who defects obtains b (b > 1) if the partner is cooperating. In other cases, he or she gets noting instead.
[29–32], group selection [33,34], tolerance [35], altruism punishment [36–38], spatially structured populations [8,39–44], social values [13,45], selective investment [3,9,46], social diversity [2,4,7,9,47], volunteerism [37,48–50], aspirations [51,52], multilayer networks [53–55], and complex networks [56,57]. Apart from them, the reputation mechanism is commonly introduced as a necessary countervailing mechanism against temptation [5,6,13,50,58–61]. Some researchers investigated the effects of reputation on the individual partner-switching process, and it indicates that the group’s cooperation propensity will increase by almost 100% if individuals are free to alter actions or partners [6]. However, some researchers argue that individuals’ reputation cannot be fully identified in reality, due to the cost and error of information dissemination [5,59]. Individuals have limited information and heterogeneous capabilities to identify the neighbors’ reputation correctly. Thus, the term of inferring reputation probability is applied to refer to the situation that each agent infers the neighbor∗ with highest reputation correctly merely with a probability p, which is normally distributed. Outcomes show that inferring reputation promotes cooperation more than traditional ways [5,58]. Although we have already known that temptation reduces cooperation substantially [5,6,38,58,59], two issues stay unclear: First, how different levels of p influence the cooperation propensity, i.e. p’s effect on cooperation rate; if reputation promotes cooperation, the second issue naturally emerges that when and where reputation effectively countervails temptation, i.e. the condition where reputation countervails temptation. The first question lays the foundation of the second one. This work applies spatial prisoners’ dilemma game to investigate and solve these issues. 2. Model Agents commonly play prisoners’ dilemma games on a square lattice [5,38] in existing researches. The prisoners’ dilemma game is also called as social dilemma game [5,57]. Agents play games with its eight neighbors, and each has two strategy options, which is cooperate and defect that are denoted as C and D respectively. Payoff matrix is shown in Fig. 1, which has only one parameter b that satisfies b ∈ (1, 2] [5,6]. If one cooperates with a neighbor who cooperates he receives one, if he or she defects with a neighbor who cooperates then the payoff is b, and the payoff would be zero otherwise. According to previous work [5,6], we utilize Eq. (1) to generate reputation for each agent. Initially, each agent is given a reputation of one. Afterwards, reputation is determined by the increment of reputation, or the strategy chosen by each agent i at each time t, i.e. i z. For each agent, i z is one if he or she cooperates and zero if he or she defects. The term Zi (t − 1) represents individual’s reputation at time t − 1, and
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Fig. 2. The action rule. Neighbor∗ refers to the neighbor with the highest reputation. For each agent, there is a referring reputation probability p, and each one imitates the action of neighbor∗ at the probability of p. If this agent chose to imitate neighbor∗ , the probability is Pij , i.e. the transition probability. One imitates a random neighbor otherwise.
Zi (t) is the accumulative current reputation of each one at time t.
Zi (t ) = Zi (t − 1) + i Z
(1)
Neighbor∗
denotes the neighbor with highest reputations. Each agent imitates his or her neighbor∗ with a probability p called inferring reputation probability [5]. In order to investigate effects of p, we assume that agents share the same level of p that satisfies p ∈ (0, 1]. Hence, as is indicated in Fig. 2, each agent finds his or her own neighbor∗ with the probability of p, and imitates action of the neighbor∗ with the transition probability Pij . In other cases, agent imitates one neighbor randomly. The formula of transition probability Pij is shown by Eq. (2), where si and sj represent agent’s current action and action of his or her neighbor∗ . At each time t or interaction, each agent chooses one specific action, either cooperation or defection, and receives a payoff. Terms ui and uj are payoffs agent i and j, respectively. The focal agent i adopts action of his or her neighbor∗ j with the transition probability Pij . Therefore, the higher the payoff of j is than that of i, the more possible it is for the focal agent to imitate action of j. Equivalently, the lower the payoff of j is than that of i, the less possible for i to imitate action of j [5,6,56].
Pi j = Psi →s j =
1 1 + e(u j −ui )β
(2)
In Eq. (2), β represents the intensity of selection, which means that β → 0 leads to random drift while β → ∞ deterministic imitation. Given that β is not our focus, we set β ≡ 1. We set the number of agents 40,000 (200 × 200) and the initial percentage or propensity of cooperation is 50%. The probability p takes on 10 typical values, i.e. p ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}. Likewise, b takes on 5 typical values from interval (1, 2], i.e. b ∈ {1.2, 1.4, 1.6, 1.8, 2.0}. For each combination of parameters, the simulation process runs for 800 iterations or periods and the cooperation propensity or percentage ρ c is recorded at each time or iteration t. 3. Reputation versus temptation There exist two mechanisms, temptation and reputation, that jointly determine the cooperation rate ρ c . The temptation mechanism means that how temptation affects cooperation. This mechanism produces a negative effect on cooperation propensity [5,6,38,58]. The reputation mechanism indicates reputation’s effect on cooperation, which is the focus of this work. We investigate absolute and relative effects of the two mechanisms on cooperation. The temptation
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Fig. 3. The temptation effect. It indicates the relationship between temptation b and rate of cooperation ρ c , and (a) represents the situation where p is no larger than 0.5 and (b) depicts the case where p > 0.5. In both (a) and (b), temptation always undermines cooperation.
mechanism is measured by parameter b, and a larger b represents a stronger temptation mechanism. Likewise, the reputation mechanism is measured by p; hence a higher p represents a greater reputation mechanism [5]. We utilize the averaged cooperation rate ρ c to measure the overall propensity of cooperation. Effects of these two mechanisms are shown as follows. 3.1. Temptation’s effect In order to inspect effects of temptation b, inferring reputation probability p is kept at the same levels. It is clear that the effect of temptation b on cooperation is clear and obvious at each level of p. It always undermines cooperation ρ c , which consists with previous studies [5,6,38,57]. Fig. 3 indicates that b reduces ρ c at any given levels of p, i.e. ρ c declines with a growing b, and a larger b comes up with a lower ρ c . When p is no larger than 0.5, this pattern is clear in Fig. 3(b). When p is larger than 0.5 or close to 1, we have to zoom in to find this pattern in Fig. 3(a). This pattern is because b seduces agents to defect other than to cooperate. As it shows in Fig. 1, individuals tend to defect if the payoff is larger than one (b > 1). The larger b is, the more agents tend to defect. In all, temptation permanently seduces people to defect as the reducing force of cooperation.
that reputation reduces cooperation when it grows closely to one. Fig. 4 tells us that there should be some type of threshold for the inferring reputation probability. When p is less than or equal to 0.5, the cooperation propensity increases, and it decrease gradually when p is larger than 0.5, especially when p is larger than 0.8. This pattern holds true for each levels of temptation. At early stages, reputation mechanism plays the role of improving cooperation, but it reduces cooperation when it increases to 1. 3.3. The joint effect It concludes that reputation and temptation jointly affect or determine the propensity of cooperation, ρ c . The temptation mechanism constantly reduces cooperation, while the reputation improves cooperation at the early stage and then reduces it when it grows close to 1. These two mechanisms jointly determine the propensity of cooperation. As is shown in Fig. 5, temptation always reduces cooperation, while reputation enhances it when p < 0.6, and reduces it when p is close to 1. Hence, it appears that reputation merely countervails temptation when p is relatively small and it does not counteract temptation when p is large. 4. The countervailing condition
3.2. Reputation’s effect Likewise, we investigate effects of reputation at same levels of temptation. Therefore, we inspect the relationship between ρ c and p at each given level of b. The effect of reputation is different from temptation, in that it enhances cooperation at first and then decreases it when close to 100%. Related work has found that reputation or inferring reputation increases cooperation [5,6,60], but it is not well revealed
The reputation countervails temptation merely when the inferring probability p is small but it cannot counteract temptation when it grows larger and even reduces cooperation as it is close to 100%. Therefore, the countervailing condition of reputation is that p should be less than 0.5. This condition is acquired by observations, and we ought to test it with a systemic or scientific way. Hence, we apply statistical methods to figure out the validity of this countervailing condition.
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Fig. 4. The reputation effect. The relationship between and the inferring reputation probability p is shown here, and we can see two thresholds here. The first threshold is about 0.5. Before 0.5, p enhances substantially, but after that it cannot do this anymore. The second threshold is about 0.9, after that p decreases.
Fig. 5. Joint effects of reputation and temptation. The combined effect of reputation (horizontal axis) and temptation (vertical axis) is shown here. Temptation has five typical values and reputation owns 10. For temptation, it reduces cooperation constantly, but reputation enhances cooperation conditionally. As p > 0.9, it decrease cooperation instead.
First we extend the simulation to get a much denser dataset. We run the simulation for 800 iterations with more parameters, i.e. p ∈ {0.025, 0.05, 0.075, 0.1, 0.125, 0.15, 0.175, 0.2, 0.225, 0.25, 0.275, 0.3, 0.325, 0.35, 0.375, 0.4, 0.425, 0.45, 0.475, 0.5, 0.525, 0.55, 0.575, 0.6, 0.625, 0.65, 0.675, 0.7, 0.725, 0.75, 0.775, 0.8, 0.825, 0.85, 0.875, 0.9, 0.925, 0.95, 0.975, 1} and b ∈ {1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0}. For each simulation, the average propensity of cooperation is recorded to indicate the overall cooperation level. We plot the new heat map with a denser dataset in Fig. 6, and the overall trend is similar to that of Fig. 5. In order to examine their effects, we calculate the conditional expectation for b and p and plot them in Fig. 6. In Fig. 6(a), temptation always undermines cooperation at its whole range, which coincides with previous work [5,6]. However, reputation promotes cooperation only at its first half domain, i.e. reputation seems to promote cooperation when p < 0.5. When p > 0.5, reputation does not enhance cooperation and even reduces cooperation while p → 1, which contradicts the previous research [5].
There seems to be threshold of p = 0.5, below which reputation promotes cooperation and beyond which cooperation does not increase cooperation. We apply statistical methods to investigate their effects and verify the threshold phenomena. Eq. (3) is applied to evaluate their effects on cooperation, and estimated coefficients represent effects or forces of reputation p and temptation b. Agents are influenced by both temptation and reputation when making choices and interactions. The dependent variable is ρ c ; coefficient β b measures effects of temptation; β p measures effects of reputation; C is constant and ɛ is the residual term.
ρc = C + βb b + β p p + ε
(3)
Method of OLS (Ordinary Least Square) estimation is applied to investigate their effects. Outcomes of OLS estima and β are OLS estitions are listed in Table 1, where β p b mations of coefficients β p and β b in Eq. (3). Coefficients are estimated under four conditions of inferring reputation probability, i.e. p ∈ (0, 1), (0, 0.5), (0.5, 1], and (0.8, 1]. For the effect
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Fig. 6. Denser joint effects. The combined effect of reputation (horizontal axis) and temptation (vertical axis) is shown here. Temptation has 10 typical values and reputation has 50, which is why this map is denser than Fig. 5. For temptation, it reduces cooperation constantly, but reputation enhances cooperation conditionally. As p > 0.9, it decreases cooperation instead.
Fig. 7. Conditional effects of reputation and temptation. It visualizes the effects of reputation and temptation based on different p to investigate the condition where reputation is able to overcome or conquer temptation to increase cooperation. When p < 0.5, the reputation overcomes temptation, but it does not when p > 0.5. Especially, when p > 0.8, both temptation and reputation reduce cooperation as their effects or coefficients are negative. Hence, reputation countervails temptation successfully merely when p is less than a half.
of temptation, it shows that temptation always reduces cooperation under all conditions of p: the overall effect of tempta(.|0 < p ≤ 1) = −0.32; conditional eftion is negative, i.e. β b (.|0 < p < 0.5) = fects of temptation are all negative, i.e. β b (.|0.8 < p ≤ 1) = (.|0.5 < p ≤ 1]) = −0.32, and β −0.69, β b b −0.02. For the effect of reputation, the overall effect on co (.|0 < p ≤ 1) = 0.53. However, operation is positive, i.e. β p its conditional effects vary a great deal: the effect is positive and reputation promotes cooperation as p is below the (.|0 < p < 0.5) = 1.32 but reputation does threshold, i.e. β p not promote cooperation as it goes beyond its threshold. The higher p is, the more it undermines cooperation, because of (.|0.5 < p ≤ 1) = −0.01 and β (.|0.8 < p ≤ 1) = −0.08. β p p
Table 1 Overall and conditional effects of reputation and temptation.
β p β b C N Adj. R2 ∗∗
p ∈ (0, 1]
p ∈ (0, 0.5)
p ∈ (0.5, 1]
p ∈ (0.8, 1]
0.53∗∗∗ (0.039) −0.32∗∗∗ (0.039) 1.08∗∗∗ (0.065) 200 0.5588
1.32∗∗∗ (0.104) −0.69∗∗∗ (0.047) 1.45∗∗∗ (0.078) 100 0.8083
−0.01∗ (0.005) −0.32∗∗∗ (0.003) 1.02∗∗∗ (0.006) 100 0.2454
−0.08∗∗∗ (0.018) −0.02∗∗∗ (0.004) 1.09∗∗∗ (0.017) 40 0.4238
p < 0.01. ∗ p < 0.05. ∗∗∗ p < 0.001.
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Table 1 shows OLS estimations of reputation and temp representing their ef and β tation, the two coefficients β p b fects respectively. Coefficients are all significant at the level of 0.001. Fig. 7 visualizes effects of temptation and reputation. The overall effect of reputation is positive (0.53), and effect of reputation is negative (−0.32). However, their effects vary with different ranges of p. In terms of reputation, when p < 0.5, the promoting force of it is strongest, which is 1.32. Nevertheless, it tends to reduce cooperation when p > 0.5, and the effect is -0.01. Furthermore, it reduces cooperation even more when p > 0.8. It is statistically significant that reputation promotes cooperation when its inferring probability is relatively lower, and tends to reduce cooperation when p > 0.5. In terms of temptation, its negative effect on cooperation decreases with p. The force analysis can be done as a metaphor. Two forces of reputation and temptation jointly influence individual choices of cooperation. Temptation plays the role of negative force to undermine cooperation all the time, and reputation plays the role of positive force to enhance cooperation conditionally. When inferring reputation probability is relatively weaker, the force of inferring reputation countervails temptation and successfully leads the cooperation to defeat defection. However, the situation changes, as p grows higher than its threshold 0.5. Inferring reputation does not promote cooperation, but reduces it together with temptation. The overall force of temptation and reputation is negative, which means that this joint force undermines cooperation when p is close to 1. This is contradictory to previous work and should arouse some further considerations. 5. Conclusions Inferring reputation resists temptation and promotes cooperation conditionally, which is not well revealed before. It was previously believed that reputation always promotes cooperation and therefore countervails temptation [5,6,58,60]. Nevertheless, it does not always do this as we wished. Actually, when the inferring reputation mechanism is weaker (p < 0.5), reputation prevents cooperation from being undermined by temptation and enhance cooperation substantially. However, when p is lager than 0.5, it does not countervail temptation, as it contributes little to cooperation or even reduces it. Therefore, it cannot stand that reputation always promotes cooperation. Although it has been verified that temptation reduces cooperation unconditionally [1,6,38], we still have new findings that the reducing force of temptation varies depending on the inferring probability p as well. The negative force of temptation decreases with the inferring probability. As is shown in Fig. 7, it is strong when p is smaller, but weak when p is larger. Previous studies believe that temptation reduces cooperation permanently, but they did not reveal how temptation affects cooperation with other parameters. There also exist two limitations. First, in order to investigate effects of reputation, it is assumed that homogenous agents share the same inferring reputation. Actually, agents or people are heterogeneous and p is usually normally distributed [5]. Other than normal distribution, there exist t-distribution, Poisson distribution, scale-free distribu-
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tion [23], multilayer networks [34,53,55], etc. These will be introduced into the future research; Second, outcomes in this paper maybe caused by individuals’ imitation process, which includes only two options, random imitation and neighbor∗ imitation. Results might be different if the third option is applied into the process. For instance, if agent can imitate himself or herself, then outcomes would be different. Acknowledgments This work is supported by the National Science Fund for Distinguished Young Scholars (grant no. 71225007), National Basic Research Program of China (973 Program) (grant no. 2011CB706900), and the National Natural Science Foundation of China (grant no. 91224008). References [1] Parks CD, Joireman J, Van Lange PAM. Cooperation, trust, and antagonism: how public goods are promoted. Psychol Sci Public Interest 2013;14:119–65. [2] Shi D-M, Yang H-X, Hu M-B, Du W-B, Wang B-H, Cao X-B. Preferential selection promotes cooperation in a spatial public goods game. Physica A 2009;388:4646–50. [3] Zhang H, Shi D, Liu R, Wang B. Dynamic allocation of investments promotes cooperation in spatial public goods game. Physica A 2012;391:2617–22. [4] Zhu C-J, Sun S-W, Wang L, Ding S, Wang J, Xia C-y. Promotion of cooperation due to diversity of players in the spatial public goods game with increasing neighborhood size. Physica A 2014;406:145–54. [5] Wang Z, Wang L, Yin Z-Y, Xia C-Y. Inferring reputation promotes the evolution of cooperation in spatial social dilemma games. PLoS ONE 2012;7:e40218. doi:10.1371/journal.pone.0040218. [6] Fu F, Hauert C, Nowak M, Wang L. Reputation-based partner choice promotes cooperation in social networks. Phys Rev E 2008;78:026117. doi:10.1103/PhysRevE.78.026117. [7] Shi D-M, Zhuang Y, Wang B-H. Group diversity promotes cooperation in the spatial public goods game. EPL 2010;90:58003. doi:10.1209/02955075/90/58003. [8] Peng D, Yang HX, Wang WX, Chen GR, Wang BH. Promotion of cooperation induced by nonuniform payoff allocation in spatial public goods game. Eur Phys J B 2010;73:455–9. [9] Li J, Wu Te, Zeng G, Wang L. Selective investment promotes cooperation in public goods game. Physica A 2012;391:3924–9. [10] Nowak MA. Five rules for the evolution of cooperation. Science 2006;314:1560–3. [11] McIntosh C, Sadoulet E, Buck S, Rosada T. Reputation in a public goods game: taking the design of credit bureaus to the lab. J Econ Behav Org 2013;95:270–85. [12] Jaramillo JJ, Srikant R. A game theory based reputation mechanism to incentivize cooperation in wireless ad hoc networks. Ad Hoc Networks 2010;8:416–29. [13] Dellarocas C. Reputation mechanism design in online trading environments with pure moral hazard. Inform Syst Res 2005;16:209–30. [14] Hofbauer J, Sigmund K. Evolutionary games and population dynamics. Cambridge, UK: Cambridge University Press; 1998. [15] Wang Z, Szolnoki A, Perc M. Percolation threshold determines the optimal population density for public cooperation. Phys Rev E 2012;85:037101. [16] Tanimoto J. Does a tag system effectively support emerging cooperation? J Theor Biol 2007;247:756–64. [17] Helbing D, Szolnoki A, Perc M, Szabó G. Evolutionary establishment of moral and double moral standards through spatial interactions. PLoS Comput Biol 2010;6:e1000758. [18] Perc M, Szolnoki A. Group-size effects on the evolution of cooperation in the spatial public goods game. Phys Rev E 2011;84:047102. [19] Wang Z, Szolnoki A, Perc M. Optimal interdependence between networks for the evolution of cooperation. Sci Rep 2013;3:2470. doi:10.1038/srep02470. [20] Poncela J, Gomez-Gardenes J, Traulsen A, Moreno Y. Evolutionary game dynamics in a growing structured population. New J Phys 2009;11:083031. [21] Perc M, Szolnoki A. Conditional strategies and the evolution of cooperation in spatial public goods games. Phys Rev E 2011;85:026104.
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